Abstract

We will discuss a new characterisation of Gromov hyperbolic groups: a group G is Gromov hyperbolic if and only if, for all rational numbers K, there are only finitely many “cone types” of (K,C)-quasigeodesics. This work is joint with Sam Hughes and Davide Spriano.

Roughly, the cone type of a quasigeodesic gamma consists of all the ways you can possibly continue gamma so that it is still a quasigeodesic.

Cannon proved in 1984 that the fundamental group of a compact hyperbolic manifold (or rather, the Cayley graph of such a group) has only finitely many cone types of geodesics. Later, in 2000, Holt and Rees generalised this result to encompass quasigeodesics: they proved that a Gromov hyperbolic group has only finitely many cone types of (K,C)-quasigeodesics provided that K is rational. Sam, Davide and I prove a strong converse to this result of Holt and Rees.